E. MAJCHRZAK1, J. MENDAKIEWICZ2
1,2 Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, 44-100 Gliwice, Konarskiego 18a
1 Institute of Mathematics and Computer Science
Czestochowa University of Technology, 42-200 Czestochowa, Dabrowskiego 73
SUMMARY
In the paper the thermal processes proceeding in the system casting -mould- environment are discussed. Heat transfer in nonhomogeneous domain considered is described by the system of partial differential equations (energy equat ions) supplemented by the adequate boundary, initial, physical and geometrical conditions.
If the parameters appearing in governing equations are known then the direct problem is considered, while if the part of them is unknown then the inverse problem sho uld be formulated. In particular in the paper presented the problem of thermal conductivity of casting material identification analyzed (the thermal conductivity of casting is assumed to be a constant value). In order to solve the inverse problem formulate d the additional information concerning the cooling curves at selected set of points from the domain considered must be given. The inverse problem is solved using the least square criterion in which the sensitivity coefficients appear. On the stage of nume rical computations the boundary element method is applied. In the final part of the paper the results of computations are shown.
Keywords: solidification process, inverse problem, parameter estimation method, boundary element method
1prof. dr hab. inż., ewa.majchrzak@polsl.pl
2dr inż., jerzy.mendakiewicz@polsl.pl
1. FORMULATION OF THE INVERS E PROBLEM
The 1D casting-mould system is considered. Transient temperature field in casting sub-domain determines the energy equation
2
1 2
( , ) ( , )
0 : ( ) T x t T x t
x L C T
t x
(1)
where C T( ) is the substitute thermal capacity [1], is the thermal conductivity, T is the temperature, x is the spatial co-ordinate and t is the time.
A temperature field in mould sub-domain describes the equation
2
1 2
( , ) ( , )
: m T x tm m T x tm
L x L c
t x
(2)
where m is the thermal conductivity and cm is the volumetric specific heat of mould.
On the contact surface between casting and mould the continuity condition
1
( , ) ( , )
:
( , ) ( , )
m m
m
T x t T x t
x x
x L
T x t T x t
(3)
is assumed. For x = 0 (axis of symmetry) the no-flux condition is accepted. For the outer surface of the system the heat transfer process is determined by condition
( , )
: m( , ) m T x tm m( , )
x L q x t T x t T
x
(4)
where α is the heat transfer coefficient, T ∞ is the ambient temperature. For the moment t = 0 the initial temperature distribution is known
0 0
( , 0) ( ) m( , 0) m ( )
T x T x T x T x (5)
Additionally, the values Td ifat the selected set of points xi from casting sub-domain for times t f are known, namely
( , ), 1, 2, .... , , 1, 2, ... ,
f f
d i d i
T T x t i M f F (6)
2. METHOD OF SOLUTION
At first, the governing equations (1) – (5) are differentiated with respect to the thermal conductivity and then
2
1 2
2
1 2
1
( , ) ( , ) ( ) ( , )
0 : ( ) λ
( , ) ( , )
: λ
( , ) ( , )
: ( , ) ( , ) ( , )
λ λ
( , )
0 : 0
( , )
: λ ( , )
0 : ( , 0) 0, ( ,
m m
m m
m
m m
m
m m
m
Z x t Z x t C T T x t
x L C T
t x t
Z x t Z x t
L x L c
t x
Z x t Z x t
x L Z x t Z x t T x t
x x x
Z x t
x x
Z x t
x L Z x t
x
t Z x Z x
0)0
(7)
where Z x t( , ) T x t( , ) / and Zm( , )x t Tm( , ) /x t are the sensitivity functions.
Next, the least squares criterion is applied [2, 3, 4]
21 1
( λ )
M F
f f
i d i
i f
S T T
(8)where Tif T x t( ,i f) is the calculated temperature at the point x for time i tf . Differentiating the criterion (8) with respect to the unknown thermal conductivity λ and using the necessary condition of minimum, one obtains
1 1 λ λ
2 0
λ λ k
M F f
f f i
i d i
i f
S T
T T
(9)where k is the number of iteration, λk for k = 0 is the arbitrary assumed value of λ, while λk for k > 0 results from the previous iteration.
Function Tif is expanded in a Taylor series about known values of λk, this means
1λ λ
( )
λ k
k if
f f k k
i i
T T T
(10)
or
k k( λ 1 λ )f f f k k
i i i
T T Z (11)
Putting (11) into (9) one has
2
1
1 1 1 1
λ λ
M F f k k k M F f k f f k
i i d i i
i f i f
Z Z T T
(12)this means
1 1
1
2
1 1
λ λ , 0, 1, ... ,
M F f k f f k
i d i i
i f
k k
M F k
f i
i f
Z T T
k K
Z
(13)This equation allows to find the values λk+1. The iteration process is stopped when the assumed accuracy is achieved.
For each iteration the basic problem and additional ones connected with t he sensitivity function have been solved using the 1st scheme of the boundary element method [5, 6] supplemented by the artificial heat source procedure [7].
3. EXAMPLE OF COMPUTATIONS
The casting-mould system of dimensions 2L1 = 0.02 [m] (casting) and 0.03 [m]
(mould) has been considered. The following input data have been introduced:
C(T) = 5.175 [MJ/m3K] for T<1470 C, C(T) = 61.4 for T[1470, 1505], C(T) = 5.74 for T>1505, m = 2.6 [W/mK], cm = 1.75 [MJ/m3K], pouring temperature T0 =1550 C, initial mould temperature Tm0 =30 C, heat transfer coefficient α = 10 [W/m2K], ambient temperature T ∞ = 30 C.
In order to identify the value of λ the courses of cooling curves (c.f. equation (7)) at the points x1 = 0 [m] (axis of symmetry), x2 = 0.004 [m] and x3 = 0.0075 [m] have been taken into account. They result from the direct problem solution under the assumption that λ = 35 [W/mK]. Figure 2 illustrates the solution of inverse problem for different initial values of λ . It is visible that the iteration process for the assumed 0 initial values of parameter λ is convergent and the solution close to the exact value is obtained after 15 iterations.
1470 1480 1490 1500 1510 1520 1530 1540 1550
0 5 10 15 20 25 t [ s ]
T [ o C ]
1
2
3
Fig. 1. Cooling curves Rys. 1. Krzywe stygnięcia
30 32 34 36 38 40
0 3 6 9 12 15 k
[ W/mK ]
Fig. 2. Estimation of Rys. 2. Estymacja parametru
REFERENCES
[1] B.Mochnacki, J.S.Suchy, Numerical methods in computations of foundry processes, PFTA, Cracow, 1995.
[2] K.Kurpisz, A.J.Nowak, Inverse thermal problems, Computational Mechanics Publications, Southampton, Boston, 1995.
[3] O.M.Alifanov, Inverse heat transfer problems, Springer-Verlag, 1994.
[4] M.N.Ozisik, H.R.B.Orlande, Inverse heat transfer: fundamentals and applications, Taylor and Francis, Pennsylvania, 1999.
[5] C.A.Brebia, J.Dominguez, Boundary elements, an introductory course,
Computational Mechanics Publications, McGraw-Hill Book Company, London, 1992.
[6] E.Majchrzak, Boundary element method in heat transfer, Publ. of Czestochowa University of Technology, Częstochowa, 2001.
[7] E.Majchrzak, B.Mochnacki, The BEM application for numerical solution of non-steady and non-linear thermal diffusion problems, Computer Assisted Mechanics and Engineering Sciences, Vol. 3, No 4, 1996, 327-346.
This paper has been sponsored by KBN (Grant No 3 T08B 004 28).
OSZACOWANIE WSPÓŁCZYNNIKA PRZEWODZENIA CIEPŁA STALIWA NA PODSTAWIE KRZYWYCH STYGNIĘCIA Z OBSZARU ODLEWU
STRESZCZENIE
W artykule analizowano procesy cieplne zachodzące w układzie odlew-forma- otoczenie. Przepływ ciepła w obszarze niejednorodnym opisuje układ równań różniczkowych (równań energii) uzupełniony odpowiednimi warunkami brzegowymi, początkowymi, fizycznymi i geometrycznymi. Jeśli parametry termofizyczne występujące w opisie matematycznym procesu są znane, wówczas rozpatruje się tzw.
zadanie bezpośrednie, natomiast, jeśli część z nich nie jest znana, to formułuje się odpowiednie zadanie odwrotne. W przedstawionym artykule zajmowano się problemem identyfikacji stałego współczynnika przewodzenia ciepła odlewu. Do rozwiązania tak sformułowanego zadania odwrotnego wykorzystano dodatkową informację dotyczącą przebiegu krzywych stygnięcia w wybranych punktach odlewu. Problem odwrotny rozwiązano stosując kryterium najmniejszych kwadratów, w którym pojawiają się współczynniki wrażliwości. Na etapie obliczeń numerycznych wykorzystano metodę elementów brzegowych. W końcowej części przedstawiono wyniki obliczeń.
Recenzował Prof. Wojciech Kapturkiewicz
